The notion of semi-exponential operators broadens the scope of classical exponential-type operators in approximation theory. In the present article, we provide a semi-exponential extension of the exponential operators connected to \(x^{4/3}\) . The kernel of these integral operators satisfy the partial differential equation, \( \frac{\partial }{\partial x}\phi _{n}^{\beta }(x,t)=\left[ \frac{n(t-x)}{x^{4/3}}-\beta \right] \phi _{n}^{\beta }(x,t)\) for \(\beta > 0\) along with normalization condition. We derive the explicit form of the moment generating function and obtain the corresponding moments and central moments. Further, we establish quantitative asymptotic formulae for these operators, along with error estimates. In addition, we provide some numerical examples alongwith graphical representation.
